TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Variation field of a geodesic variation is a Jacobi field

Statement

Let $(M, g)$ be a Riemannian manifold, let $a < b$ , let $ε > 0$ , and let $F : (- ε, ε) \times [a, b] ⟶ M$ be a smooth geodesic variation, so every longitudinal curve $t \mapsto F (s, t)$ is an affinely parametrized geodesic. Set $γ (t) = F (0, t), V (t) = {\partial_{s} F (s, t) ∣}_{s = 0} .$ Then $V$ is a Jacobi field along $γ$ : $D_{t}^{2} V + R (V, \dot{γ}) \dot{γ} = 0$ on $[a, b]$ , using one-sided derivatives at included endpoints. No fixed endpoint condition is imposed; constant geodesics and dimension zero are included. This implication uses no axiom of choice.

Facts & Assumptions

Given: The supplied Riemannian manifold and smooth geodesic variation $F : (- ε, ε) \times [a, b] \to M$ with $a < b$ .

[F1]

A geodesic variation has central geodesic $γ (t) = F (0, t)$ and variation field $V (t) = \partial_{s} F (0, t)$ ; each longitudinal curve is affinely parametrized and satisfies $D_{t} \partial_{t} F = 0$ (Geodesic variation).

[F2]

Along a smooth two-parameter map, covariant derivatives satisfy $D_{s} D_{t} W - D_{t} D_{s} W = R (F_{s}, F_{t}) W$ for every smooth field $W$ along the map (Covariant derivatives commute up to curvature in a two parameter variation).

[F3]

The Levi-Civita connection is torsion free: $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ (Levi civita connection).

[F4]

A smooth field $J$ along $γ$ is Jacobi when it satisfies $D_{t}^{2} J + R (J, \dot{γ}) \dot{γ} = 0$ (Jacobi field).

[F5]

Covariant differentiation along each parameter curve is defined by the induced pullback connection, for example $D_{t} W = (γ^{*} \nabla)_{\partial / \partial t} W$ for a curve $γ$ and a field $W$ along it (Covariant derivative along a curve).

Proof

1.1F1F5given

Put $S = F_{s}$ and $T = F_{t}$ . By [F1], $D_{t} T = 0$ on the parameter rectangle, and the central variation field is $V = S (0, \cdot)$ . Smoothness of $F$ makes $S$ and $T$ smooth fields along $F$ .

2.1F2step 1.1

Apply [F2] to the field $T$ . Since $D_{t} T = 0$ , $0 = D_{s} D_{t} T = D_{t} D_{s} T + R (S, T) T,$ so $D_{t} D_{s} T + R (S, T) T = 0$ throughout the rectangle.

3.1F2F3F4F5step 2.1

In local coordinates on $M$ , torsion freeness [F3] and equality of the mixed partial derivatives of $F$ give $D_{s} T = D_{t} S$ : the ordinary mixed derivatives agree, and the connection terms cancel because the torsion is zero. Thus [F2] and [F3] imply $D_{t}^{2} S + R (S, T) T = 0$ for every $s$ . At $s = 0$ , $S = V$ and $T = \dot{γ}$ , so [F4] shows that $V$ is Jacobi along $γ$ .

4.1F1F4F5step 3.1∎

The equation holds on the interior and extends to included endpoints by smoothness up to $t = a, b$ and the one-sided covariant derivatives. If $V = 0$ , the equation is immediate. If the central geodesic is constant, $T (0, t) = 0$ and the same computation reduces to $D_{t}^{2} V = 0$ . Empty $M$ has no supplied variation; in dimension zero every field along $F$ is zero; in dimension one the same calculation applies. The map $F$ and its derivatives are supplied, and the proof uses no selection. This is a one-way implication, not an iff claim.

Depends on

Used by

Conjugacy is a property of two points independent of the geodesic between them False statement
Local length comparison for a conjugate-free geodesic Lemma
At a conjugate endpoint the index form is degenerate Proposition
Killing fields restrict to Jacobi fields along geodesics Proposition
Differential of the exponential map in terms of Jacobi fields Theorem
Every Jacobi field is induced by a geodesic variation Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (2025) (standard reference, not scraped)